Remark 84.15.7 (Ringed variant over an object). Let $\mathcal{C}$ be a site. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings on $\mathcal{C}$. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and denote $\mathcal{O}_ X = \mathcal{O}_\mathcal {C}|_{\mathcal{C}/U}$. Then we can combine the constructions given in Remarks 84.15.5 and 84.15.6 to get

a ringed site $(\mathcal{C}/K, \mathcal{O}_ K)$ for $K$ in $\text{SR}(\mathcal{C}, X)$,

a decomposition $\textit{Mod}(\mathcal{O}_ K) = \prod \textit{Mod}(\mathcal{O}_{U_ i})$ if $K = \{ U_ i\} $,

a localization morphism $j : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K), \mathcal{O}_ K) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X), \mathcal{O}_ X)$ of ringed topoi,

a morphism $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K), \mathcal{O}_ K) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/L), \mathcal{O}_ L)$ of ringed topoi for $f : K \to L$ in $\text{SR}(\mathcal{C}, X)$.

Of course all of the results mentioned in Remark 84.15.6 hold in this setting as well.

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